IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 22, NO. 2, FEBRUARY 2013 679 Juxtaposed Color Halftoning Relying on Discrete Lines Vahid Babaei and Roger D. Hersch, Member, IEEE Abstract— Most halftoning techniques allow screen dots to in color space [3]–[5], multi-color dithering [6], juxtaposed overlap. They rely on the assumption that the inks are transpar- halftoning using screen libraries [7] and error diffusion of ent, i.e., the inks do not scatter a significant portion of the light Neugebauer primaries [8], [9]. Ostromoukhov and Hersch [6] back to the air. However, many special effect inks, such as metallic inks, iridescent inks, or pigmented inks, are not transparent. presented a juxtaposed multi-color dithering technique where In order to create halftone images, halftone dots formed by amounts of colorants are converted into dither value intervals. such inks should be juxtaposed, i.e., printed side by side. We The resulting colorant surfaces form colorant rings that follow propose an efficient juxtaposed color halftoning technique for the level lines of the dither function. By using a space filling placing any desired number of colorant layers side by side curve [10] as dither function, partially clustered juxtaposed without overlapping. The method uses a monochrome library of screen elements made of discrete lines with rational thicknesses. screen shapes can be generated [11]. Within the framework of Discrete line juxtaposed color halftoning is performed efficiently fluorescent imaging, Hersch et al. [7] created a new clustered- by multiple accesses to the screen element library. dot juxtaposed halftoning algorithm for printing images with Index Terms— Color halftoning, discrete lines, juxtaposed fluorescent inks. The resulting halftone screens allow three halftoning, Neugebauer primaries, opaque inks. colorants to be printed side by side within a given screen tile. Moroviˇc et al. [8], [9] proposed an approach for printing with freely chosen amounts of Neugebauer primaries by relying on I. INTRODUCTION error diffusion halftoning. Color separation is performed by ALFTONING algorithms try to reproduce the visual segmenting the color space into convex subspaces (e.g. tetrahe- Himpression of a continuous tone image by taking advan- dra) whose vertices are formed by the Neugebauer primaries. tage of the low-pass filtering property of the human visual Ideally, juxtaposed halftoning should have similar properties system (HVS). In classical color halftoning algorithms, such as conventional halftoning. It should provide the possibility as clustered dot and blue noise dithering, a halftone layer is of printing with a sufficient number of colorants and tone created for each ink separately [1]. The final color halftone variations. It should also provide some clustering behavior, image is formed by the superposition of all the layers. The be able to reproduce image details at a frequency higher than screen dot layers form partially overlapping screen dots. Over- the screen frequency, exhibit as least artifacts as possible and lapped screen dots form new colorants1 under the assumption offer support for an efficient implementation [12]. that the inks are transparent, i.e. they do not scatter light back In this paper, we introduce a new juxtaposed halftoning to the surface. There are however applications with strongly algorithm which creates side by side laid out colorant halftone scattering inks such as opaque inks, metallic inks, or inks lines without limitation in the number of colorants. The providing special effects such as daylight fluorescent inks proposed method relies on discrete line geometry which pro- and iridescent inks. In such applications, in order to obtain vides subpixel precision for creating discrete parallelograms. predictable halftone colors, one needs to print the different The screen elements are formed by parallelograms made of colorant halftone dots side by side without overlapping. discrete line segments whose relative subpixel thicknesses Previous attempts for side by side printing of colorants com- are set according to the desired colorant surface coverages. prise Kueppers approach of 7 color printing [2], error diffusion The parallelogram screen elements form a library comprising all possible discrete line thickness variations. The final color Manuscript received December 5, 2011; revised July 25, 2012; accepted halftone screen is created by accessing and combining binary August 29, 2012. Date of publication October 2, 2012; date of current version screen elements stored within the library. In Section II, we January 10, 2013. This work was supported in part by the Swiss National Science Foundation, under Grant 200020_126757/1. The associate editor introduce the discrete line which is the building block of our coordinating the review of this manuscript and approving it for publication juxtaposed halftoning algorithm. In Section III, we describe was Dr. Jesus Malo. the discrete line drawing algorithm. Section IV outlines the The authors are with the Ecole Polytechnique Fédérale de Lausanne, Lau- sanne 1015, Switzerland (e-mail: vahid.babaei@epfl.ch; rd.hersch@epfl.ch). procedure of creating bilevel screen elements. Multi-colorant Color versions of one or more of the figures in this paper are available juxtaposed halftoning and its efficient implementation are pre- online at http://ieeexplore.ieee.org. sented in Section V. In Section VI, we discuss the parameters Digital Object Identifier 10.1109/TIP.2012.2221727 1We use the term “colorant” for unprinted paper, solid inks and the influencing the properties of the resulting halftones. Finally, superposition of solid inks printed on paper. Classical halftones made with in Section VII we show experimental results. cyan, magenta and yellow inks comprise 8 colorants, also called Neugebauer primaries: paper white, cyan, magenta, yellow, blue as the superposition of II. DISCRETE LINES cyan and magenta, green as the superposition of cyan and yellow, red as the superposition of magenta and yellow, and black as the superposition of cyan, The arithmetic definition of a discrete line introduced magenta and yellow. by Reveillès is a fundamental notion in digital geometry 1057–7149/$31.00 © 2012 IEEE 680 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 22, NO. 2, FEBRUARY 2013 Fig. 2. Illustration of the b-periodicity of discrete lines. A discrete line with a slope equal to a/b repeats the same structure every b pixels, here a = 4and b = 7. In this work, pixels are represented by unit squares centered on integer points. A discrete line with 0 < |a| < |b| has two Euclidean support lines. The superior support line is given by a γ y = x − (3) sup b b and the inferior support line is given by a γ + w y = x − . (4) inf b b By subtracting these two equations, we obtain vertical thick- ness w/b which is the vertical distance between the superior and inferior support lines. The arithmetic thickness parameter w controls the vertical thickness and the connectivity of the line: 1) If w<|b|, the line is disconnected and we call it thin line, 2) If w =|b|, the line is strictly 8-connected; it is called naive digital line and has exactly the vertical thickness of 1, 3) If w>|b|, the line is a thick line with thickness greater than 1. Fig. 1 shows discrete lines having different thicknesses. Another interesting property of a discrete line is its b-periodicity. As shown in Fig. 2, for a given naive digital Fig. 1. Discrete lines with a = 4, b = 7, and γ =−3 with different line in the first octant with parameters a and b,afterb pixels thicknesses. (a) Thin w = 4. (b) Naive w = 7. (c) Thick w = 17. in the horizontal direction, the same line segment is repeated. Therefore, the discrete line is invariant under the translation [13]–[15]. It allows mastering the creation of discrete lines k[ab]T , for any integer k. The main advantage of b-periodicity of subpixel thicknesses. Since it is based on rational numbers, is that we can limit our study to pixels x [0, b − 1]. the discrete line plotting function does not propagate errors as Furthermore, we can use this property for efficient discrete is the case with floating point algorithms. A set D of points line plotting. (x, y) in Z2 belongs to the discrete line if and only if each member of this set satisfies III. DISCRETE LINE PLOTTING γ ≤ ax − by <γ+ w. (1) The first step towards discrete line halftoning is the ability of generating discrete lines with any desired rational thickness In other words and orientation. Due to the symmetry properties of discrete D(a, b,γ,w)= (x, y) ∈ Z2|γ ≤ ax − by <γ+ w (2) lines, without loss of generality, we limit our study to the first octant where 0 < a < b with a and b being mutually prime.2 γ w / where parameters a, b, and are integers, a b is the line 2 γ Horizontal, vertical and 45° oriented discrete lines can have only thick- slope, defines the affine offset indicating the line position in nesses in integer steps. They are therefore not usable for discrete line the plane and w determines its thickness. juxtaposed halftoning. BABAEI AND HERSCH: JUXTAPOSED COLOR HALFTONING RELYING ON DISCRETE LINES 681 Algorithm 1 Incremental Plotting of a Naive Line When plotting a thin or thick discrete line of a thickness y = Div (a ∗ x − γ,b) Integer division other than unity, instead of directly plotting the discrete line ( , ,γ,w) r = Rem (a ∗ x − γ,b) Integer remainder D a b , we synthesize a top and a bottom naive line , ,γ, ) , ,γ , ) for x = 0tox = b − 1do with parameters (a b b and (a b new b respectively, Plot Pixel (x, y) such that γ = γ + r = r + a new bt (11) if r >= b where t is the vertical thickness of desired discrete line. The r = r − b plotted thin or thick discrete line is composed of pixels with y = y + 1 pixel centers located between these two naive lines.